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Sinc-Galerkin method for numerical solution of the Bratu’s problems

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Abstract

Study of the performance of the Galerkin method using sinc basis functions for solving Bratu’s problem is presented. Error analysis of the presented method is given. The method is applied to two test examples. By considering the maximum absolute errors in the solutions at the sinc grid points are tabulated in tables for different choices of step size. We conclude that the Sinc-Galerkin method converges to the exact solution rapidly, with order, \(O(\exp{(-c \sqrt{n}}))\) accuracy, where c is independent of n.

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References

  1. Buckmire, R.: Application of a Mickens finite-difference scheme to the cylindrical Bratu-Gelfand problem. Numer. Methods Partial Differential Equations 20(3), 327–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Jacobsen, J., Schmitt, K.: The Liouville-Bratu-Gelfand problem for radial operators. J. Differential Equations 184, 283–298 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Mcgough, J.S.: Numerical continuation and the Gelfand problem. Appl. Math. Comput. 89, 225–239 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Mounim, A.S., de Dormale, B.M.: From the fitting techniques to accurate schemes for the Liouville-Bratu-Gelfand problem. Numer. Methods Partial Differential Equations 22(4), 761–775 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boyd, J.P.: Chebyshev polynomial expantions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation. Appl. Math. Comput. 142, 189–200 (2003)

    Article  MathSciNet  Google Scholar 

  6. Frank-Kamenetski, D.A.: Diffusion and Heat Exchange in Chemical Kinetics. Princeton University Press, Princeton, NJ (1995)

    Google Scholar 

  7. Caglar, H., Caglar, N., Özer, M., Anagnostopoulos, A.N.: B-spline method for solving Bratu’ problem. J. Comput. Math. 87(8), 1885–1891 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Deeba, E., Khuri, S.A., Xie, S.: An algorithm for solving boundary value problems. J. Comput. Phys. 159, 125–138 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Khuri, S.A.: A new approach to Bratu’s problem. Appl. Math. Comput. 147, 131–136 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wazwaz, A.M.: Adomian decomposition method for a reliable treatment of the Bratu-type equations. Appl. Math. Comput. 166, 652–663 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Syam, M.I., Hamdan, A.: An efficient method for solving Bratu equations. Appl. Math. Comput. 176, 704–713 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, S., Liao, S.J.: An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl. Math. Comput. 169, 854–865 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Aregbesola, Y.A.S.: Numerical solution of Bratu problem using the method of weighted residual. Electron. J. South Afr. Math. Sci. 3(1), 1–7 (2003)

    Google Scholar 

  14. Hassan, I.H.A.H., Erturk, V.S.: Applying differential transformation method to the one-dimensional planar Bratu problem. Int. J. Contemp. Math. Sci. 2, 1493–1504 (2007)

    MathSciNet  MATH  Google Scholar 

  15. He, J.H.: Some asymptotic methods for strongly nonlinear equations. Internat. J. Modern Phys. B 20(10), 1141–1199 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. He, J.H.: Variational approach to the Bratu’s problem. J. Phys.: Conf. Ser. 96, 012087 (2008)

    Article  Google Scholar 

  17. Mohsen, A., Sedeek, L.F., Mohamed, S.A.: New smoother to enhance multigridbased methods for Bratu problem. Appl. Math. Comput. 204, 325–339 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Batiha, B.: Numerical solution of Bratu-type equations by the variational iteration method. Hacet. J. Math. Stat. 39, 23–29 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Stenger, F.: A Sinc-Galerkin method of solution of boundary value problems. J. Math. Comput. 33, 85–109 (1979)

    MathSciNet  MATH  Google Scholar 

  20. Stenger, F.: Handbook of Sinc Numerical Methods. CRC Press (2010)

  21. Lund, J.: Symmetrization of the Sinc-Galerkin method for boundary value problems. J. Math. Comput. 47, 571–588 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. El-Gamel, M., Zayed, A.I.: Sinc-Galerkin method for solving nonlinear boundary value problems. J. Comput. Math. Appl. 48, 1285–1298 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mohsen, A., El-Gamel, M.: On the Galerkin and collocation methods for two-point boundary value problems using sinc bases. J. Comput. Math. Appl. 56, 930–941 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stenger, F.: Numerical Method Based on Sinc and Analytic Functions. Springer, New York (1993)

    Book  Google Scholar 

  25. Lund, J., Bowers, K.: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia, PA (1992)

    Book  MATH  Google Scholar 

  26. Stenger, F.: Summary of sinc numerical methods. J. Comput. Appl. Math. 121, 379–42 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

    Jalil Rashidinia, Khosro Maleknejad & Narges Taheri

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  1. Jalil Rashidinia
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  2. Khosro Maleknejad
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  3. Narges Taheri
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Correspondence to Jalil Rashidinia.

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Rashidinia, J., Maleknejad, K. & Taheri, N. Sinc-Galerkin method for numerical solution of the Bratu’s problems. Numer Algor 62, 1–11 (2013). https://doi.org/10.1007/s11075-012-9560-3

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  • DOI: https://doi.org/10.1007/s11075-012-9560-3

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